False starts question

We started sleep training our 4.5 month old just over a week ago and have had excellent results so far! Really happy overall with how it’s going.

Today was a busy day and unfortunately her schedule was a bit off as a result and I think she went to bed slightly under tired. This resulted in a false start 45 mins after she went down and she was completely wide awake for this. There was quite a lot of crying for 10 mins but thankfully, she put herself back to sleep in 20 minutes.

My question is what do you do if your baby wakes and becomes so wakeful that they can’t go back to sleep quickly? Our room where she sleeps is blacked out with constant white noise so it would be clear to her that it’s still bedtime but what if the sleep pressure isn’t there? Would I still follow extinction until she falls asleep and just accept it will take a long time? Thank you!